From the Icosahedron to E8

TL;DR

This paper presents two novel constructions of the E8 lattice derived from the icosahedron, connecting various mathematical objects like quaternions, the golden ratio, and singularity resolutions.

Contribution

It introduces two distinct methods to construct the E8 lattice from the icosahedron, linking diverse areas of mathematics.

Findings

Two constructions of E8 from the icosahedron are described.

Connections established between quaternions, the golden ratio, and singularity theory.

The paper challenges readers to find the link between the two constructions.

Abstract

The regular icosahedron is connected to many exceptional objects in mathematics. Here we describe two constructions of the $\mathrm{E}_8$ lattice from the icosahedron. One uses a subring of the quaternions called the "icosians", while the other uses du Val's work on the resolution of Kleinian singularities. Together they link the golden ratio, the quaternions, the quintic equation, the 600-cell, and the Poincare homology 3-sphere. We leave it as a challenge to the reader to find the connection between these two constructions.